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ThueMorse
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ThueMorse

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gives the n^(th) term in the Thue–Morse sequence.

Details

  • ThueMorse[n] is 1 exactly when n has an odd number of 1s in its binary expansion.
  • ThueMorse gives the sequence generated by the substitution starting from .
  • ThueMorse[0] is the beginning of the Thue–Morse sequence.
  • ThueMorse automatically threads over lists.

Examples

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Basic Examples  (2)Summary of the most common use cases

The fifth element of the Thue–Morse sequence:

In[1]:=1
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https://wolfram.com/xid/0yv6yb0ew1q2-kfeirw
Out[1]=1

Five has an even number of ones in its binary expansion:

In[2]:=2
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https://wolfram.com/xid/0yv6yb0ew1q2-vxg0t0

The first 10 elements of the sequence:

In[1]:=1
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https://wolfram.com/xid/0yv6yb0ew1q2-zg42
Out[1]=1

Display the values alongside the binary expansion:

In[2]:=2
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https://wolfram.com/xid/0yv6yb0ew1q2-dheyx7
Out[2]=2

Scope  (2)Survey of the scope of standard use cases

ThueMorse threads over lists:

In[1]:=1
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https://wolfram.com/xid/0yv6yb0ew1q2-0yv40m
Out[1]=1

Evaluate at large integers:

In[1]:=1
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https://wolfram.com/xid/0yv6yb0ew1q2-uk5jk6
Out[1]=1

Applications  (1)Sample problems that can be solved with this function

The Koch snowflake:

In[1]:=1
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https://wolfram.com/xid/0yv6yb0ew1q2-1l1z68
Out[1]=1

Properties & Relations  (8)Properties of the function, and connections to other functions

ThueMorse[n] is 1 if and only if n has an odd number of 1s in its binary form:

In[1]:=1
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https://wolfram.com/xid/0yv6yb0ew1q2-yjubap
Out[1]=1

The Thue–Morse sequence can arise from a nested list:

In[1]:=1
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https://wolfram.com/xid/0yv6yb0ew1q2-uo0s5
Out[1]=1

The Thue–Morse sequence can arise from the center column of a cellular automaton:

In[1]:=1
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https://wolfram.com/xid/0yv6yb0ew1q2-tqy5c
Out[1]=1

The Thue–Morse sequence has a closed form as a hypergeometric function:

In[1]:=1
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https://wolfram.com/xid/0yv6yb0ew1q2-b8z0ai
Out[1]=1

Solution to a recurrence relation:

In[1]:=1
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https://wolfram.com/xid/0yv6yb0ew1q2-bmwfjp
Out[1]=1

Cube-free sequence:

In[1]:=1
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https://wolfram.com/xid/0yv6yb0ew1q2-es34pg
In[2]:=2
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https://wolfram.com/xid/0yv6yb0ew1q2-nnapuk
In[3]:=3
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https://wolfram.com/xid/0yv6yb0ew1q2-dr3xrs
Out[3]=3

Overlap-free sequence:

In[1]:=1
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https://wolfram.com/xid/0yv6yb0ew1q2-etn0zt
In[2]:=2
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https://wolfram.com/xid/0yv6yb0ew1q2-x5c3
In[3]:=3
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https://wolfram.com/xid/0yv6yb0ew1q2-cr451p
Out[3]=3

The Thue–Morse sequence has a nested structure:

In[1]:=1
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https://wolfram.com/xid/0yv6yb0ew1q2-hrz58z
Out[1]=1

Neat Examples  (1)Surprising or curious use cases

Paths at different angles based on Thue–Morse sequences:

In[1]:=1
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https://wolfram.com/xid/0yv6yb0ew1q2-o5x971
Out[1]=1
Wolfram Research (2015), ThueMorse, Wolfram Language function, https://reference.wolfram.com/language/ref/ThueMorse.html.
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Wolfram Research (2015), ThueMorse, Wolfram Language function, https://reference.wolfram.com/language/ref/ThueMorse.html.

Text

Wolfram Research (2015), ThueMorse, Wolfram Language function, https://reference.wolfram.com/language/ref/ThueMorse.html.

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Wolfram Research (2015), ThueMorse, Wolfram Language function, https://reference.wolfram.com/language/ref/ThueMorse.html.

CMS

Wolfram Language. 2015. "ThueMorse." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/ThueMorse.html.

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Wolfram Language. 2015. "ThueMorse." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/ThueMorse.html.

APA

Wolfram Language. (2015). ThueMorse. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/ThueMorse.html

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Wolfram Language. (2015). ThueMorse. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/ThueMorse.html

BibTeX

@misc{reference.wolfram_2025_thuemorse, author="Wolfram Research", title="{ThueMorse}", year="2015", howpublished="\url{https://reference.wolfram.com/language/ref/ThueMorse.html}", note=[Accessed: 29-April-2025 ]}

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@misc{reference.wolfram_2025_thuemorse, author="Wolfram Research", title="{ThueMorse}", year="2015", howpublished="\url{https://reference.wolfram.com/language/ref/ThueMorse.html}", note=[Accessed: 29-April-2025 ]}

BibLaTeX

@online{reference.wolfram_2025_thuemorse, organization={Wolfram Research}, title={ThueMorse}, year={2015}, url={https://reference.wolfram.com/language/ref/ThueMorse.html}, note=[Accessed: 29-April-2025 ]}

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@online{reference.wolfram_2025_thuemorse, organization={Wolfram Research}, title={ThueMorse}, year={2015}, url={https://reference.wolfram.com/language/ref/ThueMorse.html}, note=[Accessed: 29-April-2025 ]}